cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A165824 Totally multiplicative sequence with a(p) = 3.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 3, 27, 9, 9, 3, 27, 3, 9, 9, 81, 3, 27, 3, 27, 9, 9, 3, 81, 9, 9, 27, 27, 3, 27, 3, 243, 9, 9, 9, 81, 3, 9, 9, 81, 3, 27, 3, 27, 27, 9, 3, 243, 9, 27, 9, 27, 3, 81, 9, 81, 9, 9, 3, 81, 3, 9, 27, 729, 9, 27, 3, 27, 9, 27, 3, 243, 3, 9, 27, 27
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2009

Keywords

Links

Crossrefs

Cf. A000244, A001222, A061142, A347149, A350961 (partial sums).

Programs

  • Maple
    A165824 := proc(n)
    3^numtheory[bigomega](n) ;
end proc:
seq(A165824(n),n=1..40) ; # R. J. Mathar, Mar 07 2022
  • Mathematica
    3^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)
  • PARI
    a(n) = 3^bigomega(n); \\ Altug Alkan, Apr 09 2016
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, 1/(1-3*X))[n], ", ")) \\ Vaclav Kotesovec, May 08 2025

Formula

a(n) = A000244(A001222(n)) = 3^bigomega(n) = 3^A001222(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 3 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

Extensions

More terms from Vaclav Kotesovec, Feb 16 2022

A069205 a(n) = Sum_{k=1..n} 2^bigomega(k).

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 25, 29, 33, 35, 43, 45, 49, 53, 69, 71, 79, 81, 89, 93, 97, 99, 115, 119, 123, 131, 139, 141, 149, 151, 183, 187, 191, 195, 211, 213, 217, 221, 237, 239, 247, 249, 257, 265, 269, 271, 303, 307, 315, 319, 327, 329, 345, 349, 365, 369, 373
Offset: 1

Views

Author

Benoit Cloitre, Apr 14 2002

Keywords

Comments

Partial sums of A061142. - Michel Marcus, Aug 08 2017

References

  • G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
  • G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 53, exercise 5 (in the third edition 2015, p. 59, exercise 57).

Links

Crossrefs

Cf. A061142, A167864, A347195, A350961.

Programs

  • Mathematica
    Accumulate[2^PrimeOmega[Range[60]]] (* Harvey P. Dale, Aug 22 2011 *)
  • PARI
    a(n) = sum(k=1, n, 2^bigomega(k)); \\ Michel Marcus, Aug 08 2017

Formula

Asymptotic formula: a(n) = 1/(8*log(2))*C*n*log(n)^2+O(n*log(n)) with C = A167864 = Product_{p primes > 2} (1+1/p/(p-2)) where the product is over all the primes p>2.
From Daniel Suteu, May 23 2020: (Start)
a(n) = Sum_{k=1..n} 2^(bigomega(k) - omega(k)) * floor(n/k).
a(n) = Sum_{k=1..n} A335073(floor(n/k)).
a(n) = 1 + Sum_{k=1..floor(log_2(n))} 2^k * pi_k(n), where pi_k(n) is the number of k-almost primes <= n. (End)
More precise asymptotics [Grosswald, 1956]: a(n) ~ A167864*n*log(n)*(log(n) - 2 - 4*A347195 + 4*gamma + 5*log(2))/(8*log(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 22 2021
Even more precise formula: a(n) ~ A167864 * n / (8*log(2)) * (log(n)^2 + (4*g + 5*log(2) - 2 - 4*A347195)*log(n) + 2 + 2*g^2 - 4*sg1 - 5*log(2) + 13*log(2)^2/6 + 2*g*(5*log(2) - 2) - 2*A347195*(5*log(2) - 2 + 4*g) + 4*A347195^2 + c), where c = Sum_{prime p > 2} (2*p * (2*p-3)* log(p)^2) / ((p-2)^2 * (p-1)^2) = 8.86809160013722347937514407919207620377461987744681170588044228288988578547..., g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Feb 11 2022

A069212 a(n) = Sum_{k=1..n} 3^omega(k).

Original entry on oeis.org

1, 4, 7, 10, 13, 22, 25, 28, 31, 40, 43, 52, 55, 64, 73, 76, 79, 88, 91, 100, 109, 118, 121, 130, 133, 142, 145, 154, 157, 184, 187, 190, 199, 208, 217, 226, 229, 238, 247, 256, 259, 286, 289, 298, 307, 316, 319, 328, 331, 340, 349, 358, 361, 370, 379, 388, 397
Offset: 1

Views

Author

Benoit Cloitre, Apr 14 2002

Keywords

Comments

More generally, if b is an integer =>3, Sum_{k=1..n} b^omega(k) ~ C(b)*n*log(n)^(b-1) where C(b)=1/(b-1)!*prod((1-1/p)^(b-1)*(1+(b-1)/p)).

References

  • G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
  • G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. (2015). See page 59.

Links

Crossrefs

Partial sums of A074816.
Cf. A001222, A064608, A065473, A350961.

Programs

  • Mathematica
    Accumulate @ Table[3^PrimeNu[n], {n, 1, 57}] (* Amiram Eldar, May 24 2020 *)
  • Python
    from sympy.ntheory.factor_ import primenu
def A069212(n): return sum(3**primenu(m) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

Formula

Asymptotic formula: a(n) ~ C*n*log(n)^2 with C = (1/2) * Product_{p} ((1-1/p)^2*(1+2/p)) where the product is over all the primes.
The constant C is A065473/2. - Amiram Eldar, May 24 2020
From Ridouane Oudra, Jan 01 2021: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} mu(i*j)^2*floor(n/(i*j));
a(n) = Sum_{i=1..n} mu(i)^2*tau(i)*floor(n/i);
a(n) = Sum_{i=1..n} 2^Omega(i)*mu(i)^2*floor(n/i), where Omega = A001222. (End)
From Vaclav Kotesovec, Feb 16 2022: (Start)
More precise asymptotics:
Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)), then
a(n) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2),
where f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927127898384464343318440970569956414778593366522...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = 0.8023233847630974628467999132875783526536954420333140745016349208975965...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} -6*p*(2*p+1) * log(p)^2 / (p^2 + p - 2)^2 = -0.255987592484328884627082229528266165335336670389046663124468278519...
and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
Showing 1-3 of 3 results.

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